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# Review, PID controller

## 1.

Review, PID ControllerMd Hazrat Ali

Department of Mechanical Engineering

School of Engineering,

Nazarbayev University

## 2.

Today’s Quote:“Live as if you were to die tomorrow. Learn as if you were to

live forever.”

― Mahatma Gandhi

## 3. Steady State Error (ess)

Steady-state error is defined as the difference between the input(command) and the output of a system in the limit as time goes to

infinity (i.e. when the response has reached steady state). The

steady-state error will depend on the type of input (step, ramp, etc.)

as well as the system type (0, I, or II).

Note: Steady-state error analysis is only useful for stable

systems. You should always check the system for stability before

performing a steady-state error analysis.

## 4. Steady State Error (ess)

## 5.

Steady State Error (ess)- Multipleinputs

## 6.

Classical ControllerPID Controller## 7.

IntroductionMore than half of the industrial controllers in use today utilize

PID or modified PID control schemes.

When the mathematical model of the plant is not known and

therefore analytical design methods cannot be used, PID

controls prove to be most useful.

Design PID control

Know mathematical model various design techniques

Plant is complicated, can’t obtain mathematical model

experimental approaches to the tuning of PID controllers

## 8. PID Control

PID ControlA closed loop (feedback) control system, generally with

Single InputSingle Output (SISO)

A portion of the signal being fed back is:

Proportional to the signal (P)

Proportional to integral of the signal (I)

Proportional to the derivative of the signal (D)

## 9. When PID Control is Used

PID control works well on SISO systems of 2nd Order, where adesired Set Point can be supplied to the system control input

PID control handles step changes well to the Set Point

especially when :

Fast Rise Times

Little or No Overshoot

Fast settling Times

Zero Steady State Error

PID controllers are often fine tuned on-site, using established

guidelines

## 10. Output equation of PID controller in time domain

## 11.

Proportional ControlA proportional controller attempts to perform better than the On-off type by applying

power in proportion to the difference in temperature between the measured and the setpoint.

The P-controller usually has steady-state errors (the difference in set point and actual

outcome) unless the control gain is large.

As the control gain becomes larger, issues arise with the stability of the feedback loop.

## 12.

Integral ControlO

u

t

p

u

t

Time

## 13.

Proportional-Integral ControlThe combination of proportional and integral terms is important to

increase the speed of the response.

Eliminate the steady state error.

O

u

t

p

u

t

Time

## 14.

## 15.

Tips for Designing a PID Controller1.

Obtain an open-loop response and determine what needs to be improved

2.

Add a proportional control to improve the rise time

3.

Add a derivative control to improve the overshoot

4.

Add an integral control to eliminate the steady-state error

5.

Adjust each of Kp, Ki, and Kd until you obtain a desired overall

response.

Lastly, please keep in mind that you do not need to implement all three controllers

(proportional, derivative, and integral) into a single system, if not necessary. For

example, if a PI controller gives a good enough response (like the above

example), then you don't need to implement derivative controller to the system.

Keep the controller as simple as possible.

## 16.

PID Controller(Conti… )

The Characteristics of P, I, and D controllers

A proportional controller (Kp) will have the effect of reducing the

rise time and will reduce, but never eliminate, the steady-state

error.

An integral control (Ki) will have the effect of eliminating the

steady-state error, but it may make the transient response worse.

A derivative control (Kd) will have the effect of increasing the

stability of the system, reducing the overshoot, and improving the

transient response.

## 17.

PID Controller(Conti… )

Proportional Control

By only employing proportional control, a steady state error

occurs.

Proportional and Integral Control

The response becomes more oscillatory and needs longer to

settle, the error disappears.

Proportional, Integral and Derivative Control

All design specifications can be reached.

## 18.

PID Controller (Conti… )Tips for Designing a PID Controller

1.

Obtain an open-loop response and determine what needs to be improved

2.

Add a proportional control to improve the rise time

3.

Add a derivative control to improve the overshoot

4.

Add an integral control to eliminate the steady-state error

5.

Adjust each of Kp, Ki, and Kd until you obtain a desired overall

response.

Lastly, please keep in mind that you do not need to implement all three controllers

(proportional, derivative, and integral) into a single system, if not necessary. For

example, if a PI controller gives a good enough response (like the above

example), then you don't need to implement derivative controller to the system.

Keep the controller as simple as possible.

## 19.

PID Controller(Conti… )

Open-Loop Control - Example

G( s )

1

2

s 10s 20

num=1;

den=[1 10 20];

step(num,den)

## 20.

PID Controller(Conti… )

Proportional Control - Example

The proportional controller (Kp) reduces the rise time, increases

the overshoot, and reduces the steady-state error.

Kp

T( s )

MATLAB Example

2

s 10 s ( 20 Kp )

Step Response

From: U(1)

1.4

Kp=300;

1.2

num=[Kp];

step(num,den,t)

0.9

0.8

0.8

0.7

0.6

0.4

K=300

0.2

0.6

To: Y(1)

Amplitude

t=0:0.01:2;

From: U(1)

1

To: Y(1)

den=[1 10 20+Kp];

Step Response

1

Amplitude

0.5

K=100

0.4

0.3

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.2

2

Time (sec.)

0.1

0

0

0.2

0.4

0.6

0.8

1

Time (sec.)

1.2

1.4

1.6

1.8

2

## 21.

PID Controller(Conti… )

Proportional - Derivative - Example

The derivative controller (Kd) reduces both the overshoot and the

settling time.

Kd s Kp

T( s )

MATLAB Example

2

s ( 10 Kd ) s ( 20 Kp )

Step Response

From: U(1)

1.4

1.2

Kp=300;

From: U(1)

To: Y(1)

0.9

0.6

0.8

0.7

Kd=10

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

Time (sec.)

1.2

1.4

1.6

0.6

To: Y(1)

den=[1 10+Kd 20+Kp];

0.8

Amplitude

num=[Kd Kp];

t=0:0.01:2;

Step Response

1

Amplitude

Kd=10;

1

0.5

0.4

1.8

0.3

2

Kd=20

0.2

step(num,den,t)

0.1

0

0

0.2

0.4

0.6

0.8

1

Time (sec.)

1.2

1.4

1.6

1.8

2

## 22.

PID Controller(Conti… )

Proportional - Integral - Example

The integral controller (Ki) decreases the rise time, increases both

the overshoot and the settling time, and eliminates the steady-state

error

Kp s Ki

T( s )

3

MATLAB Example

2

s 10 s ( 20 Kp ) s Ki

Step Response

From: U(1)

1.4

Kp=30;

1.2

Step Response

t=0:0.01:2;

step(num,den,t)

1.2

0.8

1

Ki=70

0.4

To: Y(1)

0.6

A m plitude

den=[1 10 20+Kp Ki];

To: Y(1)

num=[Kp Ki];

Amplitude

1

Ki=70;

0.2

0

From: U(1)

1.4

0.8

0.6

0.4

0

0.2

0.4

0.6

0.8

1

Time (sec.)

1.2

1.4

1.6

1.8

Ki=100

2

0.2

0

0

0.2

0.4

0.6

0.8

1

Time (sec.)

1.2

1.4

1.6

1.8

2

## 23.

PID Controller(Conti… )

The Characteristics of P, I, and D controllers

CL RESPONSE

RISE TIME

OVERSHOOT

SETTLING TIME

S-S ERROR

Kp

Decrease

Increase

Small Change

Decrease

Ki

Decrease

Increase

Increase

Eliminate

Kd

Small Change

Decrease

Decrease

Small Change

## 24. Figure 4.9 Responses of P, PI, and PID control to (a) step disturbance input (b) step reference input

PID Controller(Conti… )

Figure 4.9 Responses of P, PI, and PID control to (a) step disturbance input (b) step reference input

## 25. Figure 4.10 Model of a satellite attitude control: (a) basic system; (b) PD control; (c) PID control

PID Controller(Conti… )

Figure 4.10 Model of a satellite attitude control: (a) basic system; (b) PD control; (c) PID control

## 26. Figure 4.11 Process reaction curve

PID ControllerFigure 4.11 Process reaction curve

## 27.

Figure 4.11 Process reaction curves (R.C.Dorf et.al and Others)## 28. Figure 4.12 Quarter decay ratio

PID Controller- Ziegler Method #1Figure 4.12 Quarter decay ratio

## 29. TABLE 4.2

PID ControllerTABLE 4.2

(Conti… )

## 30. Figure 4.13 Determination of ultimate gain and period

PID Controller- Ziegler Method #2Figure 4.13 Determination of ultimate gain and period

## 31. Figure 4.14 Neutrally stable system

PID ControllerFigure 4.14 Neutrally stable system

(Conti… )

## 32. TABLE 4.3

PID ControllerTABLE 4.3

(Conti… )

## 33. Further Reading

Franklin, et. al., Chapter 4Section 4.3

Richard C. Dorf et.al, Chapter 6,

Chapter 6.2